How to Select a Project Team: Calculating Combinations and Probabilities

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To select a project team of five staff from three team leaders and ten programmers, with the requirement of at least three programmers and one team leader, various combinations must be calculated. For the IT consultant, the probability of obtaining both contracts A and B can be found by multiplying their individual probabilities, resulting in 0.18. The probability of obtaining at least one contract involves calculating the complementary probability of not getting either contract, yielding a result of 0.78. These problems illustrate fundamental concepts in permutations, combinations, and probability theory. Understanding these calculations is essential for effective project team selection and contract bidding strategies.
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please help me if you have the time, thanks in advance :)

Question 1 :

An IT department has three team leaders and ten programmers. A project team consisting of Five staff is to be selected and it must include at least three programmers and one team leader. How many different ways can the team be selected?

Question 2:

An IT consultant bids for two independant contracts with local companies. The consultant estimates that the probability that he obtains contract A is 0.6 and the probability that he obtains contract B is 0.3. What is the probability :

i) he obtains both contracts?
ii) he obtains at least one contract?
 
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Please post your attempt at solving the problem. These are basic problems in Permutations and Combinations/Probability.
 
And since I can't imagine wanting to solve such a problem except for homework, I am moving it.
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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